Gaussian quadrature involving Einstein and Fermi functions with an application to summation of series
Authors:
Walter Gautschi and Gradimir V. Milovanović
Journal:
Math. Comp. 44 (1985), 177190
MSC:
Primary 65D32; Secondary 33A65, 65A05, 65B10, 8108, 8208
DOI:
https://doi.org/10.1090/S00255718198507710391
MathSciNet review:
771039
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Abstract: Polynomials ${\pi _k}( \cdot ) = {\pi _k}( \cdot ;d\lambda )$, $k = 0,1,2, \ldots$, are constructed which are orthogonal with respect to the weight distributions $d\lambda (t) = {(t/({e^t}  1))^r}\;dt$ and $d\lambda (t) = {(1/({e^t} + 1))^r}\;dt$, $r = 1,2$, on $(0,\infty )$. Momentrelated methods being inadequate, a discretized Stieltjes procedure is used to generate the coefficients ${\alpha _k},{\beta _k}$ in the recursion formula ${\pi _{k + 1}}(t) = (t  {\alpha _k}){\pi _k}(t)  {\beta _k}{\pi _{k  1}}(t)$, $k = 0,1,2, \ldots$, ${\pi _0}(t) = 1$, ${\pi _{  1}}(t) = 0$. The discretization is effected by the GaussLaguerre and a composite Fejér quadrature rule, respectively. Numerical values of ${\alpha _k},{\beta _k}$, as well as associated error constants, are provided for $0 \leqslant k \leqslant 39$. These allow the construction of Gaussian quadrature formulae, including error terms, with up to 40 points. Examples of npoint formulae, $n = 5(5)40$, are provided in the supplements section at the end of this issue. Such quadrature formulae may prove useful in solid state physics calculations and can also be applied to sum slowly convergent series.

J. S. Blakemore, Solid State Physics, 2nd ed., Saunders, Philadelphia, Pa., 1974.
W. Gautschi, "Algorithm 542—Incomplete gamma functions," ACM Trans. Math. Software, v. 5, 1979, pp. 482489.
 Walter Gautschi, A survey of GaussChristoffel quadrature formulae, E. B. Christoffel (Aachen/Monschau, 1979) Birkhäuser, BaselBoston, Mass., 1981, pp. 72–147. MR 661060
 Walter Gautschi, On generating orthogonal polynomials, SIAM J. Sci. Statist. Comput. 3 (1982), no. 3, 289–317. MR 667829, DOI https://doi.org/10.1137/0903018 J. F. Hart et al., Computer Approximations, Wiley, New YorkLondonSydney, 1968. A. McLellan IV, "Tables of the Riemann zeta function and related functions," Math. Comp., v. 22, 1968, Review 69, pp. 687688. S. S. Mitra & N. E. Massa, "Lattice vibrations in semiconductors," Chapter 3 in: Band Theory and Transport Properties (W. Paul, ed.), pp. 81192. Handbook on Semiconductors (T. S. Moss, ed.), Vol. 1. NorthHolland, Amsterdam, 1982. F. Reif, Fundamentals of Statistical and Thermal Physics, McGrawHill, New York, 1965. R. A. Smith, Semiconductors, 2nd ed., Cambridge Univ. Press, Cambridge, 1978. R. J. Van Overstraeten, H. J. DeMan & R. P. Mertens, "Transport equations in heavy doped silicon," IEEE Trans. Electron Dev., v. ED20, 1973, pp. 290298.
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© Copyright 1985
American Mathematical Society